DOCSLIB.ORG

Comprehensive List of Mathematical Symbols

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Comprehensive List of Mathematical Symbols Comprehensive List of Mathematical Symbols Comprehensive List of Mathematical Symbols Comprehensive List of Mathematical Symbols For the corresponding web guides, see Mathematical Symbols. Table of Contents 1 Constant ......................... 3 1.1 Key Mathematical Numbers ........... 3 1.2 Key Mathematical Sets ............. 4 1.3 Key Mathematical Infinities ........... 5 1.4 Other Key Mathematical Objects ........ 6 2 Variables ......................... 6 2.1 Variables for Numbers .............. 6 2.2 Variables in Geometry ............. 7 2.3 Variables in Calculus .............. 7 2.4 Variables in Linear Algebra ........... 8 2.5 Variables in Set Theory and Logic ....... 8 2.6 Variables in Probability and Statistics ...... 9 3 Delimiters ........................ 10 3.1 Common Delimiters ............... 10 3.2 Other Delimiters ................ 10 4 Operators ......................... 11 2 Table of Contents Comprehensive List of Mathematical Symbols 4.1 Common Operators ............... 11 4.2 Function-related Operators ........... 12 4.3 Elementary Functions .............. 12 4.4 Algebra-related Operators ............ 13 4.5 Geometry-related Operators .......... 14 4.6 Logic-related Operators ............. 15 4.7 Set-related Operators .............. 16 4.8 Vector-related Operators ............ 16 4.9 Matrix-related Operators ............ 17 4.10 Probability-related Operators .......... 18 4.11 Statistics-related Operators ........... 18 4.12 Key Probability Functions and Distributions .. 19 4.13 Calculus-related Operators ........... 20 5 Relational Symbols .................... 21 5.1 Equality-based Relational Symbols ....... 21 5.2 Comparison-based Relational Symbols ..... 21 5.3 Number-related Relational Symbols ...... 22 5.4 Geometry-related Relational Symbols ..... 22 5.5 Set-related Relational Symbols ......... 22 5.6 Logic-related Relational Symbols ........ 23 5.7 Probability-related Relational Symbols ..... 23 5.8 Calculus-related Relational Symbols ...... 24 6 Notational Symbols ................... 24 6.1 Common Notational Symbols .......... 24 6.2 Notational Symbols in Geometry and Trigonom- etry ........................ 25 6.3 Notational Symbols in Calculus ......... 25 6.4 Notational Symbols in Probability and Statistics 26 7 Additional Resources .................. 26 1 Constant 1.1 Key Mathematical Numbers 1.1 Key Mathematical Numbers 3 Comprehensive List of Mathematical Symbols Symbols LaTeX Code Example (Explanation) 0 $0$ 3 + 0 = 3 (Zero, additive identity) 1 $1$ 5 × 1 = 5 (One, multiplicative identity) √ √ √ 2 $\sqrt{2}$ ( 2 + 1)2 = 3 + 2 2 (Square root of 2) e $e$ ln(e2) = 2 (Euler’s constant) 2 π 1 1 ··· π $\pi$ = 2 + 2 + (Pi, Archimedes’ 6 1 2 constant) √ 1 + 5 φ $\varphi$ φ = (Phi, golden ratio) 2 i $i$ (1 + i)2 = 2i (Imaginary unit) 1.2 Key Mathematical Sets Symbols LaTeX Code Example (Explanation) ? $\varnothing$ |?| = 0 (Empty set) N $\mathbb{N}$ ∀x, y ∈ N, x + y ∈ N (Set of natural numbers) Z $\mathbb{Z}$ N ⊆ Z (Set of integers) 4 1.2 Key Mathematical Sets Comprehensive List of Mathematical Symbols Z+ $\mathbb{Z}_+$ 3 ∈ Z+ (Set of positive integers) √ Q $\mathbb{Q}$ 2 ∈/ Q (Set of rational numbers) R $\mathbb{R}$ ∀x ∈ R (x2 ≥ 0) (Set of real numbers) R+ $\mathbb{R}_+$ ∀x, y ∈ R+ (xy ∈ R+) (Set of positive real numbers) C $\mathbb{C}$ ∃z ∈ C (z2 + 1 = 0) (Set of complex numbers) Zn $\mathbb{Z}_n$ In the world of Z2, (Set of integer modulo 1 + 1 = 0. n) R3 $\mathbb{R}^3$ (5, 1, 2) ∈ R3 (Three-dimensional Euclidean space) 1.3 Key Mathematical Infinities Symbols LaTeX Code Example (Explanation) ℵ0 $\aleph_0$ ℵ0 + 5 = ℵ0 (Cardinality of natural numbers) ℵ c $\mathfrak{c}$ c = 2 0 (Cardinality of real numbers) ω $\omega$ ∀n ∈ N (n < ω) (Smallest infinite ordinal number) 1.3 Key Mathematical Infinities 5 Comprehensive List of Mathematical Symbols 1.4 Other Key Mathematical Objects Symbols LaTeX Code Example (Explanation) 0 $\mathbf{0}$ ∀v ∈ V, v + 0 = v (Zero vector) e $e$ e ◦ e = e (Identity element of a group) I $I$ AI = IA = A (Identity matrix) Z C $C$ 1 dx = x + C (Constant of integration) > $\top$ For each proposition P , (Tautology) P ∧ > ≡ P . ⊥ $\bot$ For each proposition P , (Contradiction) P ∧ ¬P ≡ ⊥. Z $Z$ Z ∼ N(0, 1) (Standard normal distribution) 2 Variables 2.1 Variables for Numbers Symbols LaTeX Code Example (Explanation) m, n, p, q $m$, $n$, $p$, $q$ m + n − q = 1 (Integers and natural numbers) 6 2.1 Variables for Numbers Comprehensive List of Mathematical Symbols a, b, c $a$, $b$, $c$ ax + by = 0 (Coefficients for functions and equations) x, y, z $x$, $y$, $z$ If 2x + 5 = 3, then (Unknowns in x = −1. functions and equations) ∆ $\Delta$ ∆ = b2 − 4ac for (Discriminant) quadratic polynomials X10 i, j, k $i$, $j$, $k$ i = 55 (Index variables) i=1 t $t$ At t = 5, the velocity (Time variable) is v(5) = 32. z $z$ zz¯ = |z|2 (Complex numbers) 2.2 Variables in Geometry Symbols LaTeX Code Example (Explanation) P , Q, R, S $P$, $Q$, $R$, $S$ PQ ⊥ QR (Vertices) ℓ $\ell$ ℓ1 k ℓ2 (Lines) α, β, γ, θ $\alpha$, $\beta$, α + β + θ = 180◦ (Angles) $\gamma$, $\theta$ 2.3 Variables in Calculus Symbols LaTeX Code Example (Explanation) 2.3 Variables in Calculus 7 Comprehensive List of Mathematical Symbols f(x), g(x, y), h(z) $f(x)$, $g(x,y)$, f(2) = g(3, 1) + 5 (Functions) $h(z)$ 3 an, bn, cn $a_n$, $b_n$, an = (Sequences) $c_n$ n + 2 eh − e0 h, ∆x $h$, $\Delta x$ lim = 1 → (Limiting variables in h 0 h derivatives) δ, ε $\delta$, For all ε > 0, there is a (Small quantities in $\varepsilon$ δ > 0 such that |x| < δ proofs involving implies that |2x| < ε. limits) F (x), G(x) $F(x)$, $G(x)$ F 0(x) = f(x) (Antiderivatives) 2.4 Variables in Linear Algebra Symbols LaTeX Code Example (Explanation) u, v, w $\mathbf{u}$, 3u + 4v = w (Vectors) $\mathbf{v}$, $\mathbf{w}$ A, B, C $A$, $B$, $C$ AX = B (Matrices) λ $\lambda$ Av = λv (Eigenvalues) 2.5 Variables in Set Theory and Logic Symbols LaTeX Code Example (Explanation) A, B, C $A$, $B$, $C$ A ⊆ B ∪ C (Sets) 8 2.5 Variables in Set Theory and Logic Comprehensive List of Mathematical Symbols a, b, c $a$, $b$, $c$ a ∈ A (Elements) P , Q, R $P$, $Q$, $R$ P ∨ ¬P ≡ > (Propositions) 2.6 Variables in Probability and Statistics Symbols LaTeX Code Example (Explanation) X, Y , Z $X$, $Y$, $Z$ E(X + Y ) = (Random variables) E(X) + E(Y ) µ $\mu$ H0 : µ = 5 (Population means) σ $\sigma$ σ1 = σ2 (Population standard deviations) s $s$ s =6 σ (Sample standard deviations) n $n$ For n ≥ 30, use the (Sample sizes) normal distribution. ρ $\rho$ Ha : ρ < 0 (Population correlations) r $r$ If r = 0.75, then (Sample correlations) r2 = 0.5625. π $\pi$ π = 0.5 (Population proportions) X p $p$ p = (Sample proportions) n 2.6 Variables in Probability and Statistics 9 Comprehensive List of Mathematical Symbols 3 Delimiters 3.1 Common Delimiters Symbols LaTeX Code Example (Explanation) . $.$ 25.9703 (Decimal separator) : $:$ 1 : 4 : 9 = 3 : 12 : 27 (Ratio indicator) , $,$ (3, 5, 12) (Object separator) (), [], {} $()$, $[]$, $\{ \}$ (a + b) × c (Order-of-operation indicators) (), [] $()$, $[]$ 3 ∈/ (3, 4], 4 ∈ (3, 4] (Interval indicators) 3.2 Other Delimiters Symbols LaTeX Code Example (Explanation) a 1 4 (), [], x y , $()$, $[]$, b $\begin{pmatrix} x 3 6 (Vector/matrix & y \end{pmatrix}$, indicators) $\begin{bmatrix} a \\ b \end{bmatrix}$ {} $\{ \}$ {π, e, i} (Set builder) |, : $\mid, :$ {x ∈ R | x2 − 2 = 0} (“Such that” markers) 10 3.2 Other Delimiters Comprehensive List of Mathematical Symbols ||, kk $| |, \| \|$ k(3, 4)k = 5 (Metric-related operators) f(x) x ≥ a 1 x ≥ 0 $\begin{cases} f(x) f(x) = g(x) x < a & x \ge a \\ g(x) & 0 x < 0 (Piecewise-function x < a \end{cases}$ marker) hi $\langle \rangle$ hka, bi = kha, bi (Inner product operator) de $\lceil \rceil$ d2.476e = 3 (Ceiling operator) bc $\lfloor \rfloor$ bπc = 3 (Floor operator) 4 Operators 4.1 Common Operators Symbols LaTeX Code Example (Explanation) x + y $x+y$ 2a + 3a = 5a (Sum) x − y $x-y$ 11 − 5 = 6 (Difference) −x $-x$ −3 + 3 = 0 (Additive inverse) x × y, x · y, xy $x \times y$, (m + 1)n = mn + n (Product) $x \cdot y$, $xy$ x ÷ y, x/y $x \div y$, $x/y$ 152 ÷ 3 = 50.6 (Quotient) 4.1 Common Operators 11 Comprehensive List of Mathematical Symbols x 53 + 5 53 5 $\displaystyle = + y \frac{x}{y}$ 6 6 6 (Fraction) xy $x^y$ 34 = 81 (Power) √ −b ± ∆ x ± y $x \pm y$ (Plus and minus) 2a √ √ x $\sqrt{x}$ 2 ≈ 1.414 (Positive square root) |x| $|x|$ |x − 3| < 5 (Absolute value) . x x% $x \%$ x% = (Percent) 100 4.2 Function-related Operators Symbols LaTeX Code Example (Explanation) dom f $\operatorname{dom}f$ If g(x) = ln x, then (Domain) dom(g) = R. ran f $\operatorname{ran}f$ If h(y) = sin y, then (Range) ran(h) = [−1.1]. f(x) $f(x)$ g(5) = g(4) + 3 (Image of an element) f(X) $f(X)$ f(A∩B) ⊆ f(A)∩f(B) (Image of a set) f ◦ g $f \circ g$ If g(3) = 5 and f(5) = (Composite 8, then (f ◦ g)(3) = 8. function) 4.3 Elementary Functions 12 4.3 Elementary Functions Comprehensive List of Mathematical Symbols Symbols LaTeX Code Example (Explanation) n 0 knx + ··· + k0x $k_n x^n + \cdots The polynomial (Polynomial) + k_0x^0$ x3 + 2x2 + 3 has a root in (−3, −2). ex, exp x $e^x$, $\exp x$ ex+y = ex · ey (Natural exponential function) bx $b^x$ 2x > x2 for large x. (General exponential function) ln x $\ln x$ ln(x2) = 2 ln x (Natural logarithmic function) log x $\log x$ log 10000 = 4 (Common logarithmic function) ln x logb x $\log_b x$ log2 x = (General logarithmic ln 2 function) sin x $\sin x$ sin π = 0 (Sine function) √ π 2 cos x $\cos x$ cos = (Cosine function) 4 2 sin x tan x $\tan x$ tan x = (Tangent function) cos x 4.4 Algebra-related Operators Symbols LaTeX Code Example (Explanation) gcd(x, y) $\gcd (x,y)$ gcd(35, 14) = 7 (Greatest common factor) 4.4 Algebra-related Operators 13 Comprehensive List of Mathematical Symbols bxc $\lfloor x \rfloor$ b3.6c = 3 (Floor operator) dxe $\lceil x \rceil$ dπe = 4 (Ceiling operator) min(A) $\min (A)$ If min(A) = 3, then (Minimum) min(A + 5) = 8.
Load more

Recommended publications
  • Section 3.6 Complex Zeros
    210 Chapter 3 Section 3.6 Complex Zeros When finding the zeros of polynomials, at some point you're faced with the problem x 2 −= 1. While there are clearly no real numbers that are solutions to this equation, leaving things there has a certain feel of incompleteness. To address that, we will need utilize the imaginary unit, i. Imaginary Number i The most basic complex number is i, defined to be i = −1 , commonly called an imaginary number . Any real multiple of i is also an imaginary number. Example 1 Simplify − 9 . We can separate − 9 as 9 −1. We can take the square root of 9, and write the square root of -1 as i. − 9 = 9 −1 = 3i A complex number is the sum of a real number and an imaginary number. Complex Number A complex number is a number z = a + bi , where a and b are real numbers a is the real part of the complex number b is the imaginary part of the complex number i = −1 Arithmetic on Complex Numbers Before we dive into the more complicated uses of complex numbers, let’s make sure we remember the basic arithmetic involved. To add or subtract complex numbers, we simply add the like terms, combining the real parts and combining the imaginary parts. 3.6 Complex Zeros 211 Example 3 Add 3 − 4i and 2 + 5i . Adding 3( − i)4 + 2( + i)5 , we add the real parts and the imaginary parts 3 + 2 − 4i + 5i 5 + i Try it Now 1. Subtract 2 + 5i from 3 − 4i .
    [Show full text]
  • Chapter 4 Complex Numbers Course Number
    Chapter 4 Complex Numbers Course Number Section 4.1 Complex Numbers Instructor Objective: In this lesson you learned how to perform operations with Date complex numbers. Important Vocabulary Define each term or concept. Complex numbers The set of numbers obtained by adding real number to real multiples of the imaginary unit i. Complex conjugates A pair of complex numbers of the form a + bi and a – bi. I. The Imaginary Unit i (Page 328) What you should learn How to use the imaginary Mathematicians created an expanded system of numbers using unit i to write complex the imaginary unit i, defined as i = Ö - 1 , because . numbers there is no real number x that can be squared to produce - 1. By definition, i2 = - 1 . For the complex number a + bi, if b = 0, the number a + bi = a is a(n) real number . If b ¹ 0, the number a + bi is a(n) imaginary number .If a = 0, the number a + bi = bi is a(n) pure imaginary number . The set of complex numbers consists of the set of real numbers and the set of imaginary numbers . Two complex numbers a + bi and c + di, written in standard form, are equal to each other if . and only if a = c and b = d. II. Operations with Complex Numbers (Pages 329-330) What you should learn How to add, subtract, and To add two complex numbers, . add the real parts and the multiply complex imaginary parts of the numbers separately. numbers Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company.
    [Show full text]
  • CHAPTER 8. COMPLEX NUMBERS Why Do We Need Complex Numbers? First of All, a Simple Algebraic Equation Like X2 = −1 May Not Have
    CHAPTER 8. COMPLEX NUMBERS Why do we need complex numbers? First of all, a simple algebraic equation like x2 = 1 may not have a real solution. − Introducing complex numbers validates the so called fundamental theorem of algebra: every polynomial with a positive degree has a root. However, the usefulness of complex numbers is much beyond such simple applications. Nowadays, complex numbers and complex functions have been developed into a rich theory called complex analysis and be- come a power tool for answering many extremely difficult questions in mathematics and theoretical physics, and also finds its usefulness in many areas of engineering and com- munication technology. For example, a famous result called the prime number theorem, which was conjectured by Gauss in 1849, and defied efforts of many great mathematicians, was finally proven by Hadamard and de la Vall´ee Poussin in 1896 by using the complex theory developed at that time. A widely quoted statement by Jacques Hadamard says: “The shortest path between two truths in the real domain passes through the complex domain”. The basic idea for complex numbers is to introduce a symbol i, called the imaginary unit, which satisfies i2 = 1. − In doing so, x2 = 1 turns out to have a solution, namely x = i; (actually, there − is another solution, namely x = i). We remark that, sometimes in the mathematical − literature, for convenience or merely following tradition, an incorrect expression with correct understanding is used, such as writing √ 1 for i so that we can reserve the − letter i for other purposes. But we try to avoid incorrect usage as much as possible.
    [Show full text]
  • Lecture 15: Section 2.4 Complex Numbers Imaginary Unit Complex
    Lecture 15: Section 2.4 Complex Numbers Imaginary unit Complex numbers Operations with complex numbers Complex conjugate Rationalize the denominator Square roots of negative numbers Complex solutions of quadratic equations L15 - 1 Consider the equation x2 = −1. Def. The imaginary unit, i, is the number such that p i2 = −1 or i = −1 Power of i p i1 = −1 = i i2 = −1 i3 = i4 = i5 = i6 = i7 = i8 = Therefore, every integer power of i can be written as i; −1; −i; 1. In general, divide the exponent by 4 and rewrite: ex. 1) i85 2) (−i)85 3) i100 4) (−i)−18 L15 - 2 Def. Complex numbers are numbers of the form a + bi, where a and b are real numbers. a is the real part and b is the imaginary part of the complex number a + bi. a + bi is called the standard form of a complex number. ex. Write the number −5 as a complex number in standard form. NOTE: The set of real numbers is a subset of the set of complex numbers. If b = 0, the number a + 0i = a is a real number. If a = 0, the number 0 + bi = bi, is called a pure imaginary number. Equality of Complex Numbers a + bi = c + di if and only if L15 - 3 Operations with Complex Numbers Sum: (a + bi) + (c + di) = Difference: (a + bi) − (c + di) = Multiplication: (a + bi)(c + di) = NOTE: Use the distributive property (FOIL) and remember that i2 = −1. ex. Write in standard form: 1) 3(2 − 5i) − (4 − 6i) 2) (2 + 3i)(4 + 5i) L15 - 4 Complex Conjugates Def.
    [Show full text]
  • Notes on Euler's Work on Divergent Factorial Series and Their Associated
    Indian J. Pure Appl. Math., 41(1): 39-66, February 2010 °c Indian National Science Academy NOTES ON EULER’S WORK ON DIVERGENT FACTORIAL SERIES AND THEIR ASSOCIATED CONTINUED FRACTIONS Trond Digernes¤ and V. S. Varadarajan¤¤ ¤University of Trondheim, Trondheim, Norway e-mail: [email protected] ¤¤University of California, Los Angeles, CA, USA e-mail: [email protected] Abstract Factorial series which diverge everywhere were first considered by Euler from the point of view of summing divergent series. He discovered a way to sum such series and was led to certain integrals and continued fractions. His method of summation was essentialy what we call Borel summation now. In this paper, we discuss these aspects of Euler’s work from the modern perspective. Key words Divergent series, factorial series, continued fractions, hypergeometric continued fractions, Sturmian sequences. 1. Introductory Remarks Euler was the first mathematician to develop a systematic theory of divergent se- ries. In his great 1760 paper De seriebus divergentibus [1, 2] and in his letters to Bernoulli he championed the view, which was truly revolutionary for his epoch, that one should be able to assign a numerical value to any divergent series, thus allowing the possibility of working systematically with them (see [3]). He antic- ipated by over a century the methods of summation of divergent series which are known today as the summation methods of Cesaro, Holder,¨ Abel, Euler, Borel, and so on. Eventually his views would find their proper place in the modern theory of divergent series [4]. But from the beginning Euler realized that almost none of his methods could be applied to the series X1 1 ¡ 1!x + 2!x2 ¡ 3!x3 + ::: = (¡1)nn!xn (1) n=0 40 TROND DIGERNES AND V.
    [Show full text]
  • Operations with Complex Numbers Adding & Subtracting: Combine Like Terms (풂 + 풃풊) + (풄 + 풅풊) = (풂 + 풄) + (풃 + 풅)풊 Examples: 1
    Name: __________________________________________________________ Date: _________________________ Period: _________ Chapter 2: Polynomial and Rational Functions Topic 1: Complex Numbers What is an imaginary number? What is a complex number? The imaginary unit is defined as 풊 = √−ퟏ A complex number is defined as the set of all numbers in the form of 푎 + 푏푖, where 푎 is the real component and 푏 is the coefficient of the imaginary component. An imaginary number is when the real component (푎) is zero. Checkpoint: Since 풊 = √−ퟏ Then 풊ퟐ = Operations with Complex Numbers Adding & Subtracting: Combine like terms (풂 + 풃풊) + (풄 + 풅풊) = (풂 + 풄) + (풃 + 풅)풊 Examples: 1. (5 − 11푖) + (7 + 4푖) 2. (−5 + 7푖) − (−11 − 6푖) 3. (5 − 2푖) + (3 + 3푖) 4. (2 + 6푖) − (12 − 4푖) Multiplying: Just like polynomials, use the distributive property. Then, combine like terms and simplify powers of 푖. Remember! Multiplication does not require like terms. Every term gets distributed to every term. Examples: 1. 4푖(3 − 5푖) 2. (7 − 3푖)(−2 − 5푖) 3. 7푖(2 − 9푖) 4. (5 + 4푖)(6 − 7푖) 5. (3 + 5푖)(3 − 5푖) A note about conjugates: Recall that when multiplying conjugates, the middle terms will cancel out. With complex numbers, this becomes even simpler: (풂 + 풃풊)(풂 − 풃풊) = 풂ퟐ + 풃ퟐ Try again with the shortcut: (3 + 5푖)(3 − 5푖) Dividing: Just like polynomials and rational expressions, the denominator must be a rational number. Since complex numbers include imaginary components, these are not rational numbers. To remove a complex number from the denominator, we multiply numerator and denominator by the conjugate of the Remember! You can simplify first IF factors can be canceled.
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Euler and His Work on Infinite Series
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 44, Number 4, October 2007, Pages 515–539 S 0273-0979(07)01175-5 Article electronically published on June 26, 2007 EULER AND HIS WORK ON INFINITE SERIES V. S. VARADARAJAN For the 300th anniversary of Leonhard Euler’s birth Table of contents 1. Introduction 2. Zeta values 3. Divergent series 4. Summation formula 5. Concluding remarks 1. Introduction Leonhard Euler is one of the greatest and most astounding icons in the history of science. His work, dating back to the early eighteenth century, is still with us, very much alive and generating intense interest. Like Shakespeare and Mozart, he has remained fresh and captivating because of his personality as well as his ideas and achievements in mathematics. The reasons for this phenomenon lie in his universality, his uniqueness, and the immense output he left behind in papers, correspondence, diaries, and other memorabilia. Opera Omnia [E], his collected works and correspondence, is still in the process of completion, close to eighty volumes and 31,000+ pages and counting. A volume of brief summaries of his letters runs to several hundred pages. It is hard to comprehend the prodigious energy and creativity of this man who fueled such a monumental output. Even more remarkable, and in stark contrast to men like Newton and Gauss, is the sunny and equable temperament that informed all of his work, his correspondence, and his interactions with other people, both common and scientific. It was often said of him that he did mathematics as other people breathed, effortlessly and continuously.
    [Show full text]
  • Ring (Mathematics) 1 Ring (Mathematics)
    Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right.
    [Show full text]
  • On the Euler Integral for the Positive and Negative Factorial
    On the Euler Integral for the positive and negative Factorial Tai-Choon Yoon ∗ and Yina Yoon (Dated: Dec. 13th., 2020) Abstract We reviewed the Euler integral for the factorial, Gauss’ Pi function, Legendre’s gamma function and beta function, and found that gamma function is defective in Γ(0) and Γ( x) − because they are undefined or indefinable. And we came to a conclusion that the definition of a negative factorial, that covers the domain of the negative space, is needed to the Euler integral for the factorial, as well as the Euler Y function and the Euler Z function, that supersede Legendre’s gamma function and beta function. (Subject Class: 05A10, 11S80) A. The positive factorial and the Euler Y function Leonhard Euler (1707–1783) developed a transcendental progression in 1730[1] 1, which is read xedx (1 x)n. (1) Z − f From this, Euler transformed the above by changing e to g for generalization into f x g dx (1 x)n. (2) Z − Whence, Euler set f = 1 and g = 0, and got an integral for the factorial (!) 2, dx ( lx )n, (3) Z − where l represents logarithm . This is called the Euler integral of the second kind 3, and the equation (1) is called the Euler integral of the first kind. 4, 5 Rewriting the formula (3) as follows with limitation of domain for a positive half space, 1 1 n ln dx, n 0. (4) Z0 x ≥ ∗ Electronic address: [email protected] 1 “On Transcendental progressions that is, those whose general terms cannot be given algebraically” by Leonhard Euler p.3 2 ibid.
    [Show full text]
  • 1 Sets and Set Notation. Definition 1 (Naive Definition of a Set)
    LINEAR ALGEBRA MATH 2700.006 SPRING 2013 (COHEN) LECTURE NOTES 1 Sets and Set Notation. Definition 1 (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most often name sets using capital letters, like A, B, X, Y , etc., while the elements of a set will usually be given lower-case letters, like x, y, z, v, etc. Two sets X and Y are called equal if X and Y consist of exactly the same elements. In this case we write X = Y . Example 1 (Examples of Sets). (1) Let X be the collection of all integers greater than or equal to 5 and strictly less than 10. Then X is a set, and we may write: X = f5; 6; 7; 8; 9g The above notation is an example of a set being described explicitly, i.e. just by listing out all of its elements. The set brackets {· · ·} indicate that we are talking about a set and not a number, sequence, or other mathematical object. (2) Let E be the set of all even natural numbers. We may write: E = f0; 2; 4; 6; 8; :::g This is an example of an explicity described set with infinitely many elements. The ellipsis (:::) in the above notation is used somewhat informally, but in this case its meaning, that we should \continue counting forever," is clear from the context. (3) Let Y be the collection of all real numbers greater than or equal to 5 and strictly less than 10. Recalling notation from previous math courses, we may write: Y = [5; 10) This is an example of using interval notation to describe a set.
    [Show full text]
  • Reachability Problems in Quaternion Matrix and Rotation Semigroups
    Reachability Problems in Quaternion Matrix and Rotation Semigroups Paul Bell , Igor Potapov ∗ Department of Computer Science, University of Liverpool, Ashton Building, Ashton St, Liverpool L69 3BX, U.K. Abstract We examine computational problems on quaternion matrix and rotation semigroups. It is shown that in the ultimate case of quaternion matrices, in which multiplication is still associative, most of the decision problems for matrix semigroups are un- decidable in dimension two. The geometric interpretation of matrix problems over quaternions is presented in terms of rotation problems for the 2 and 3-sphere. In particular, we show that the reachability of the rotation problem is undecidable on the 3-sphere and other rotation problems can be formulated as matrix problems over complex and hypercomplex numbers. Key words: Quaternions, Matrix semigroups, Rotation semigroups, Membership problem, Undecidability, Post’s correspondence problem 1 Introduction Quaternions have long been used in many fields including computer graphics, robotics, global navigation and quantum physics as a useful mathematical tool for formulating the composition of arbitrary spatial rotations and establishing the correctness of algorithms founded upon such compositions. Many natural questions about quaternions are quite difficult and correspond to fundamental theoretical problems in mathematics, physics and computational theory. Unit quaternions actually form a double cover of the rotation group ∗ Corresponding author. Email addresses: [email protected] (Paul Bell), [email protected] (Igor Potapov). Preprint submitted to Elsevier 4 July 2008 SO3, meaning each element of SO3 corresponds to two unit quaternions. This makes them expedient for studying rotation and angular momentum and they are particularly useful in quantum mechanics.
    [Show full text]